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Full text of "Treatise On Analysis Vol-Ii"

226 XIII INTEGRATION (13.21.9) A ^-measurable function h is v-integrable if and only if This follows from (13.21.8) and (13.9.13). (13.21.10) Let A be a ^-measurable set in X x Y. (i) The set M of all x e X such that the section A(x) is not \i-measurable is ^-negligible', the function jth-»/^*(A(.x)) is ^-measurable; and *(A) = J V In particular, ifA(x) is ^-negligible except on a ^-negligible set of values ofx, then A is ^-negligible. (ii) If A is v-integrable, then the set of all x e X such that A(x) is not \L-integrable is ^-negligible; the function x\-*/j,(A(x)) (which is defined almost everywhere with respect to X) h X-integrable; and v(A) = \fi(A(x)) dk(x). These assertions are particular cases of (13.21.6), (13.21.8) and (13.21.7). (13.21.11). Letf(resp.g) be a mapping ofX (resp. Y) into [0, +00]. With the convention of (1 3.11 ) for products, we have = ( f * \J (13.21.11.1) /(xfoOO d%x) dp(y) = f */(x) By virtue of (13.21.4), we have jj /W000 d%x) dn(y) ^ f * dl(x) Ff(x)g(y) dn(y). On the other hand, for each x e X we have ( and f*# x) J *f(x)g(y) dn(y) = f *( ( g(y)(13.21.7) the functions x^fax.y) d^(y) are A-integrable and we have